Summation formulas of hyperharmonic numbers with their generalizations

Abstract

In 1990, Spieß  gave some identities of harmonic numbers including the types \(\sum _{\ell =1}^n\ell ^k H_\ell \), \(\sum _{\ell =1}^n\ell ^k H_{n-\ell }\) and \(\sum _{\ell =1}^n\ell ^k H_\ell H_{n-\ell }\). In this paper, we derive several formulas of hyperharmonic numbers including \(\sum _{\ell =0}^{n} {\ell }^{p} h_{\ell }^{(r)} h_{n-\ell }^{(s)}\) and \(\sum _{\ell =0}^n \ell ^{p}\left( h_{\ell }^{(r)}\right) ^{2}\). Some more formulas of generalized hyperharmonic numbers are also shown.